1 Answer
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Semantically, the idiom eat one's way through means to eat, with a connotation that there is a lot of food.

It is an example of what you might call an "idiom schema" that can be used with various verbs which refer to actions one might do on a lot of individual items, again with a connotation that there are many things to deal with:

I worked my way through all the forms they presented me with.

He read his way through all the reports

or even

They sang their way through all the madrigals in the book.

But I think that eat and work are the most common verbs you will find in this schema.